A comprehensive analysis of the finite-Larmor-radius (FLR) fluid moment equations for collisionless magnetized plasmas is presented. It is based on perturbative but otherwise general solutions for the second and third rank fluid moments (the stress and stress flux tensors) with closure conditions on the fourth rank moment. The single expansion parameter is the ratio between the gyroradius of the plasma species under consideration and any other characteristic length, which is assumed to be small but finite in a magnetized medium. This formalism allows a complete account of the gyroviscous stress, the pressure anisotropy and the anisotropic heat fluxes, and is valid for arbitrary magnetic geometry, arbitrary plasma pressure and fully electromagnetic nonlinear dynamics. As the result, very general yet notably compact perturbative systems of FLR collisionless fluid equations, applicable to either fast (magnetohydrodynamic) or slow (diamagnetic) motions, are obtained.

The paraxial expansion of a collisionless plasma jet into vacuum, guided by a magnetic nozzle, is studied with an Eulerian and non-stationary Vlasov-Poisson solver. Parametric analyses varying the magnetic field expansion rate, the size of the simulation box, and the electrostatic potential fall are presented. After choosing the potential fall leading to a zero net current beam, the steady states of the simulations exhibit a quasi-neutral region followed by a downstream sheath. The latter, an unavoidable consequence of the finite size of the computational domain, does not affect the quasi-neutral region if the box size is chosen appropriately. The steady state presents a strong decay of the perpendicular temperature of the electrons, whose profile versus the inverse of the magnetic field does not depend on the expansion rate within the quasi-neutral region. As a consequence, the electron distribution function is highly anisotropic downstream. The simulations revealed that the ions reach a higher velocity during the transient than in the steady state and their distribution functions are not far from mono-energetic. The density percentage of the population of electrons trapped during the transient, which is computed self-consistently by the code, is up to 25% of the total electron density in the quasi-neutral region. It is demonstrated that the exact amount depends on the history of the system and the steady state is not unique. Nevertheless, the amount of trapped electrons is smaller than the one assumed heuristically by kinetic stationary theories.